Emmerich--Cordella Optimized Certificate for the Unit Distance Problem with Extended Prime Number Range This Zenodo record contains the certificate data and verification code for an enlarged-(T) unit-distance lower-bound certificate found by Michael T. M. Emmerich and Francesco Cordella using integer optimization (https://arxiv.org/abs/2606.03419). It is the to date sharpest (largest) lower bound certificate for a lower bound of the unit-distance problem in the certificate setting established by Sawin 2026*. The purpose of this deposit is to provide a stable, citable certificate-and-code archive. It is not intended as an arXiv addendum. A separate note by Michael T. M. Emmerich and Francesco Cordella may later discuss the enlarged (T)-range, the optimization strategy, and the mathematical context in more detail. The certificate is conceptually based on the explicit unit-distance lower-bound framework developed in the following works: @misc{emmerich2026optimizingexplicitunitdistancelowerbound, title={Optimizing Explicit Unit-Distance Lower-Bound Certificates}, author={Michael T. M. Emmerich}, year={2026}, eprint={2606.03419}, archivePrefix={arXiv}, primaryClass={math.OC}, url={https://arxiv.org/abs/2606.03419}, } with accompaning GitHub with the used integer optimization-verification pipeline: emmerichmtm/UnitDistanceProblemOptimizationOfSawinsLowerBound @article{sawin2026explicit, title={An explicit lower bound for the unit distance problem}, author={Sawin, Will}, journal={arXiv preprint arXiv:2605.20579}, year={2026} } Authors Michael Emmerich (Faculty of Information Technology, University of Jyväskylä, Finland), corresponding author. Francesco Cordella (Francesco Cordella (0000-0003-0731-5622) - ORCID) Headline certificate The deposited certificate has the following headline parameters: #T = 67 #S_Q = 1021 R = 33066458/1000000 = 33.066458 delta = 0.031185334443176590595661471677599365... split primes in Q(sqrt(prod(T))): 0 budget: 67 + 1021 + 0 + 1 = 1089 = (67-1)^2/4 verification status: passed = True Here (T) denotes the ramified prime set in the Sawin-style explicit unit-distance criterion. In this certificate, (T) consists of the first 67 odd primes, from 3 through 337. Main files cordella_certificate_data.txtCanonical compact certificate data. This is the authoritative data file for this Zenodo record. verify_certificates.pyExisting repository verification pipeline. It defines the Certificate dataclass, evaluates the exponent formula, and checks the finite arithmetic side conditions. verify_best_certificate.pyZenodo entry-point script. It parses cordella_certificate_data.txt, constructs the certificate, calls verify_certificate(...), checks the headline values, and writes JSON/TXT verification outputs. Uses the cited Emmerich 2026 (arxiv) verification pipeline. CERTIFICATE_VERIFICATION_REPORT.mdHuman-readable verification report and contextual explanation. verification_result_full.jsonFull machine-readable verification output, including the complete lists, splitting statuses, admissibility witnesses, and all checks. verification_summary.jsonCompact machine-readable summary of the certificate, suitable for quick inspection. verification_result.txtHuman-readable text output from the verification pipeline. Reproducing the verification Use Python 3.10 or newer. No third-party Python packages are required. Run: python verify_best_certificate.py or, equivalently, ./run_verification.sh Expected final output: headline checks: - T_size_is_67: True - S_Q_size_is_1021: True - R_matches_expected: True - computed_delta_matches_claimed_prefix: True - pipeline_passed: True The script exits with status code 0 only if all headline checks and all verification-pipeline checks pass. Scope of this deposit This deposit provides a stable record of the certificate data and its verification. It does not construct planar point coordinates. Rather, it verifies the finite arithmetic certificate data used in the explicit lower-bound criterion. *the certificate has not yet undergone a rigorous peer review by the mathematical community and the result needs to be treated cautiously for this reason. The authors hope that the verification effort will be useful for subsequent checks and questions can be directed to the first author via his email: michael.t.m.emmerich@jyu.fi License note No license is selected in this generated package. Before making the Zenodo record public, the authors should choose an appropriate license for the data/report and a license for the code.